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Rectangular (i) x=r cos 0 Polar Conversion Formulas (ii) y=r sin (iii) x² + y² = r² (a) Use implicit differentiation to find the slope and inclination of the tangent lines to the graph of the cardioid: (x² + y² + y)² = x² + y² at the points P = (1,0) and Q = (-1,0). (b) Use the conversion formulas (i), (ii) and (iii) noted above to rewrite the equation for the cardioid from part (a) in terms of only r and as: r = 1 - sin 0. (c) Sketch a careful graph which displays the cardioid and its tangent lines at P and Q.

Rectangular (i) x=r cos 0 Polar Conversion Formulas (ii) y=r sin (iii) x² + y² = r² (a) Use implicit differentiation to find the slope and inclination of the tangent lines to the graph of the cardioid: (x² + y² + y)² = x² + y² at the points P = (1,0) and Q = (-1,0). (b) Use the conversion formulas (i), (ii) and (iii) noted above to rewrite the equation for the cardioid from part (a) in terms of only r and as: r = 1 - sin 0. (c) Sketch a careful graph which displays the cardioid and its tangent lines at P and Q.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Rectangular (i) x=r cos 0 Polar Conversion Formulas (ii) y=r sin 0 (iii) x² + y² = r² 1. (a) Use implicit differentiation to find the slope and inclination of the tangent lines to the graph of the cardioid: (x² + y² + y)² = x² + y² at the points P = (1,0) and Q = (-1,0). (c) (b) Use the conversion formulas (i), (ii) and (iii) noted above to rewrite the equation for the cardioid from part (a) in terms of only r and as: r = 1 - sin 0. Sketch a careful graph which displays the cardioid and its tangent lines at P and Q. You should check your plot with a graphing calculator or else use the website https://www.desmos.com/calculator 2. (a) Use implicit differentiation to find the slope and inclination of the tangent line to the graph of the circle: (x - 1)² + y² = 1 at the point P = ( 1 √3 , ). 2 (b) Apply the conversion formulas (i), (ii) and (iii) to the equation from part (a) and simplify it, to obtain the polar equation of the circle: r = 2 cos 0. (c) Sketch a careful graph which displays the circle and its tangent line at P. 3. (a) Use implicit differentiation to find the slope and inclination of the tangent lines to the graph of the parabola: y² = 4x at the points P = (1,2) and Q = (4,-4). (b) Apply the conversion formulas (i), (ii) and (iii) to the equation from part (a) and simplify, to obtain the polar equation of the parabola: r = 4 cot 0.csc 0. (c) Sketch a careful graph which displays the parabola and its tangent lines at P and Q. 4. (a) Use implicit differentiation to find dy dx for the lemniscate: (x² + y²)² = x² - y². (b) Convert the equation of the lemniscate to polar form using formulas (i), (ii) and (iii). Check the accuracy of your solution by plotting it with a graphing device or desmos.com. (x² + y²)² = x² - y²
Transcribed Image Text:Rectangular (i) x=r cos 0 Polar Conversion Formulas (ii) y=r sin 0 (iii) x² + y² = r² 1. (a) Use implicit differentiation to find the slope and inclination of the tangent lines to the graph of the cardioid: (x² + y² + y)² = x² + y² at the points P = (1,0) and Q = (-1,0). (c) (b) Use the conversion formulas (i), (ii) and (iii) noted above to rewrite the equation for the cardioid from part (a) in terms of only r and as: r = 1 - sin 0. Sketch a careful graph which displays the cardioid and its tangent lines at P and Q. You should check your plot with a graphing calculator or else use the website https://www.desmos.com/calculator 2. (a) Use implicit differentiation to find the slope and inclination of the tangent line to the graph of the circle: (x - 1)² + y² = 1 at the point P = ( 1 √3 , ). 2 (b) Apply the conversion formulas (i), (ii) and (iii) to the equation from part (a) and simplify it, to obtain the polar equation of the circle: r = 2 cos 0. (c) Sketch a careful graph which displays the circle and its tangent line at P. 3. (a) Use implicit differentiation to find the slope and inclination of the tangent lines to the graph of the parabola: y² = 4x at the points P = (1,2) and Q = (4,-4). (b) Apply the conversion formulas (i), (ii) and (iii) to the equation from part (a) and simplify, to obtain the polar equation of the parabola: r = 4 cot 0.csc 0. (c) Sketch a careful graph which displays the parabola and its tangent lines at P and Q. 4. (a) Use implicit differentiation to find dy dx for the lemniscate: (x² + y²)² = x² - y². (b) Convert the equation of the lemniscate to polar form using formulas (i), (ii) and (iii). Check the accuracy of your solution by plotting it with a graphing device or desmos.com. (x² + y²)² = x² - y²
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